V .G.Soloviev

SIMPLE CONFIGURATIONS

IN THE CAPTURE STATE

AND THE GENERAL PICTURE

E4 - 8116

OF NUCLEAR STATE COMPLICATIONS

JINR PUBLICATIONS munications of the Joint Institute for tre co,nside~ed to be original publica­

of the JINR

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V .G.Soloviev

SIMPLE CONFIGURATIONS

IN THE CAPTURE STATE

AND THE GENERAL PICfURE

E4 - 8116

OF NUCLEAR STATE COMPLICATIONS

Submitted to the Second International Symposium on Neutron Capture y -Ray and Related Topics, Netherlands, 1974.

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Soloviev V.G. E4 - 8116

Simple Configurations in the Capture State and the General Picture of Nuclear State Complications

See the Abstract in the text.

Preprint. Joint Institute for Nuclear Research.

Dubna, 1974

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I

ABSTRACT

It is asserted that the effective nuclear forces and the methods of solving the ~-body problem maT serve as a basis for describing states of low, intermediate and high excitation energy. It is indicated that it is important to study nuclear structure complications with increasing excitation energy and fragmentation of single-particle and many-particle states.

The foundations of the model for describing fragmentation which is based on the account or the quasiparticle-phonon interac­tion are presented. The results of calculations of fragmentation for the 6Jll. , 640 I , 6201 and 600 I one-quasiparticle states in 23'u ~e given. It is shown that the state strength is distri­buted over a wide energy interval, the distribution maximum is shifted down with respect to the single-particle energy. The desree of fragmentation is shown to depend strongly on the position of the single-particle level with respect to the Fenn1 level. A modi­fied version of the model is given for treating highly excited sta­tes of the type of giant resonances and for studying their influen-

ce on the structure of states of intermediate excitation energy. The general assumptions of the approach based on the operator

form of the wave function of highly excited states are presented, and the contribution of individual simple configurations to the neutron resonance wave functions is estimated. It is shown in which case the valence neutron model can be employed. The compound­-state structure is discussed.

I. The general regularities of the behaviour of the excited states of complex ( medium......,eight and heavy) nuclei are rather well studied. They reflect the universal properties of the atomic nuclei and are the following:

i) In all the nuclei there are low....lying quasiparticle and vibrational collective states. In deformed nuclei they generate rotational bands. These states are rather simple, they are well studied and are correctly described in terms of quasiparticles and

1 phonons, e.g.,re£. • 11) The highly excited states are ver;r complex and are clef:l.­

ned by a large number of degrees of freedom. The highl7 excited

3

state wave functions contain thousands of various components. Random distribution of most degrees of freedom permits interpre­ting these compound states within the framework of the statistical model.

iii) Among the highly excited states are states with rather

large few-quasiparticle and one-phonon components. These are the isobar analogue states, the isospin of which is by unity higher than that of the surrounding states and the collective states which are given the name of giant resonances. There is a large group of giant resonances corresponding to different multipoles with isoscalar and isovector parts. Among tbem-thegiant El-resonan­ces are the most completely studied ones.

iv) With increasing excitation energy the level density grows and the level structure becomes more complicated. Thus, a transition proceeds from simple low-lying states to more complica­ted states with intermediate excitation energy and then to very complicated highly excited states. The structure complication proceeds in different ways in magio, vibrational, transition and deformed nuclei. However, i~ all the medium and heavy nuclei at the neutron binding energy En the states are so complicated that they are usually described within the framework of the statisti­cal model.

One has long been showing an increasing interest in the study of the structure of highly excited states. Progress is associated with the construction of the resonance reaction theory 2 and the development of neutron spectroscopy, e.g., ref.J • Of much importan­ce is also the description of highly excited states within the framework of the particle-hole approach 4 and its further develop­ment. Valuable experimental information about highly excited sta­tes is extracted from charged particles and heavy ion induced reactions. Of a special importance are electron-nucleus scattering and photonuclear reactions, e.g.,ref. 5.

One of the main universal properties of nuclei, namely compli­cation of the state structure with increasing excitation energy, is worst studied. The present report is mostly devoted to some attempm to find a general approach to the study of this problem.

The detailed study of nuclear structure shows that theoretical­ly the transition from low-lying states to states of intermediate excitation energy is predominant. However, the complex experimen­tal study of intermediate states qy means of nuclear reactions and mass separators on particle beam advances slowly. A large amount of progress has been made in the field of neutron resonan­ces. A special situation with neutron resonances is due to the

4

availability of complete and exact experimental data rather than to their physical meaning. The wave functions of neutron resonan­ces are very complex, and therefore we may only hope to find some of their few-quasiparticle components. The problem of extracting from experimental data some information about a number of simplest configurations in the neutron resonance wave functions is aLso discussed in the present report.

2. The low-lying states of complex nuclei are rather well described within the framework of the semi-microscopic approach. The Hamiltonian describing tne 1nteraction between the nucleons in the nucleus is of the form

H = Hav +Hpo.•Z "'Ho. 'H' (I)

In other words, the effective nuclear interaction is represented in the form of the average field described by the Saxon-woods po­tential, the interactions leading to superconducting pairing corre­lations, the multipole-multipole and other interactions. In the semi-mioroscopic description one calculates the relative values rather than the absolute ones, for example, the energies of excited states with respect to the ground state or changes of the nucleus energy with increasing deformation parameters.

An analysis performed in ref. 6 shows that the effective nuc­lear interactions (I) and the methods of solving nuclear many-bo­dy problem may serve as a basis for describing low, intermediate and higHWexcited states of atomic nuclei.

In the understanding of the structure of highly excited sta­tes and their description in the language of quasipartioles and phonons in which the low-lying states are also treated the main role is attributed to the process called fragmentation of fractio­nation. By fragmentation we mean the distribution of the strength of a one-, two- or many-particle states over many nuclear levels. In other words, fragmentation is responsible, for example, for the distribution of the strength of the single-particle state being the solution of the Schrodinger equation with the Saxon­Woods potential over some nuclear levels.

There are two main causes leading to complications of the sta­te structure with increasing excitation energy: the first one is the interaction of the single-particle and collective motions described as the interaction of quasi-particles with phonons; the second cause is the coupling of the intrinsic and rotational mo­tions described via the Coriolis interaction.

The quasiparticle-phonon interaction is of much importance in the process of fragmentation. It leads to mixing of the oompo-

5

nents differing by one phonon or by two quasiparticles. As a re­sult of accounting for the quasiparticle-phonon interaction, the wave function has the form of a sum of one-, two-, three- and higher phonon components in the case of doubly even nuclei.

The quasiparticle-phonon interaction strongly affects the structure of low-lying states of deformed l,7, 8 and spherical l, 9

nuclei. It results in a fragmentation of the strength of one-phonon, two-phonon and many-phonon states over nuclear levels. Some examp­les of fragmentation of the strength of subshells due to def~rma­tion and fragmentation of two-qusipartiole states caused by multi­pole-multipole phonon-generating interactions are given in ref.10 •

The development of a unified description of low, intermediate and highly excited states and the study of the process of structu­re complication with increasing excitation energy are performed by us along two lines: i) a general semi-microscopic description on the basis of the operator form of the wave function lO-lJ ; and ii) calculations on the basis of a model taking into account the interactions of quasiparticles with phonons lO,l4-l7 •

J. Let us consider a model for the description of fragmenta­tion in the case of an odd deformed nucleus. Originally in ref. 14

fragmentation was studied by the method applied for the calculati­on of the structure of low-lying states. It was also shown that this method requiressome modifications. In ref.15 a model for describing the nonrotational state structure at intermediate and high energies and for studyin~ fragmentation was formulated which was then generalized in ref.1 •

The model liamiltonian is taken in the form (I) without the term H · • The wave function of a nonrotational state with a given K7[ is wr1tten as follows

LJ ( i/11:) - {;__ ') {c-L • -t- ' o¥' 1+ (/ t I~ 1\ - v"2" ,__ P" L f""v .,_ '17 .:;;_J

C"' :f,Y (2)

+) \ rl./='. ,. 0.,. 0 .} '" ~ '-- ro>o d-. • T, JY., ~ - .a- /! "' o '

where If. - is the ground state wave function for a doubly even nucleus, i is the number of the state. The normalization condi­tion (2) is of the form

( LJ. "r K"J 'IJ K'')) = 1 = r cf'' Y{ 1 + f ~ ( DA' ;<:.,. CJ)

-.-zz= I (F';'"'/j. J"tJ, 1 p

since by introducing the phase factors 8, and Oz it is possib­le to get rid of the u dependence. We use the following notation:

6

Q1

, v..:3

and 'fs are the operator, the energy and the phonon characteristic; by 3 we mean )./1./ , j 'I ( Y, Y') is the matrix element of the multipole operator 9 = Aj.t. , j is the number of the root of the secular equation for a phonon; c/. ;,- and E ( v)

are the creation operator and the quasiparticle energy, respective-ly; r1 (YfJ = v;,,. j 'OJ!') /2 v~ , lln = Uy U/ - ~~j) ~· ,

( Ycr) and 6 = ! I is the set of quantum numbers of a single-particle level, ( prr) the same for levels with fixed K;;: •

The nonrotaUonal s~ate energies 2, and the functions c; D l' and f~J;J'' are defined iii th.e variational principle

J'' J

aU if ( K7[JH11 Lf1. (K7i))- z1 u 4f'r x!l'; 0 r K7[J J- 1 J } = o . (4)

After some transformations the system of basic equations is

written in the form

EuJ-t-L_ [(i(P,YJ Op1; =-O 9 y '

(c (J)) .. w!- P.J D.P'~- L L jJ(¥ ~n-h I o,I1'(v, Yz) r''(Y, y"' DJ'~'.

Jz Yz Y.J 1 ' " 'I; L l ,

1- o'·r1zu Y. >f'( ~ }'I D1''j = (i(p y) .( I l. & ) 2/ Pli I

F. u,t_ r "' [o.r'zc)l J ns• + (} :1-J nt•' -: J'll - j>(YJ y,y,J-2; i l ,y plz Oz (~,JJ) JlY _

(5)

(6)

(7)

The poles like fU,j1,f2 ) =c ()l)rw! •IJJt< are called funda-mental poles. To find the energies ~. it is necessary: to solve eq. (6) with respect to q,f and insert the obtained expression in eq. (5). After solving eq. (5), that is after finding the roots

i, , and using eqs. (J) and (7) it is not difficult to derive ~~~·and c; . The solution for eq. (6) can be represented

in the form

o1' =A! 1'~ A '

(8)

where the denominator A is the determinant of the system, and the numerator A! is the dete:nninant whioh is obtained from A by replacement of the coefficients for the unknown quantity by the appropriate free terms. In some real oases, when a large number of single-particle states and phonons are taken into account, eq. (8) comprises determ:tnants of the order of 10

4 and higher.

An approximate method of salTing eqs. (5) and (6) was develo-

7

ped in ref.17 • According to this method, all the coherent terms and pole noncoherent terms are taken into account. The obtained approximate equations which differ from each other by the pole terms, are solved analytically. The expressions for 4~' are inserted in eq. (5) which is then solved numerically. In each case the solution corresponding to the fundamental pole is sought. An effective method of numerical solution of the secular equations (5) was suggested there, and the secular equation derived by this method contain no superfluous solutions.

The accuracy of the approximate solution is investigated in rer.17 by the example of a restricted basis. The comparison of the wave function components for the exact and approximate solutions shows that their large components are close to one another, while very s~all components may strongly differ from one another.

The approximate method of solving the model equations is used for the study of fragmentation of the one-quasiparticle, quasi­particle plus phonon and quasiparticle plus two-phonons states over many nuclear l·evels, as well as for the study of the structu­re of the nonrotational states of intermediate excitation energy in odd deformed nuclei.

The study of the model shows that there is a direct and rather simple method of calculating the level density at different

excitation energies and different spins. It is based on the calcu­lation of the number of fundamental poles with fixed Kll: or 1"' in a given energy interval. This method was used in refs.18

t19 to

calculate the density of a large number of spherical and deformed nuclei. The account of vibrational and rotational motions has resulted in a good description of the level density near Bn.

At intermediate excitation energies, the energy and spin dependence of the level density calculated by our method may strongly differ from the statistical model calculations. For example, in 157Fe at g ~ Bn ! 0.5 MeV the density of negative parity states~ larger than that of positive parity states by a factor of 1.5 - 2.0. On the contrary, in 58re the density of the positive parity states is several times larger than that of nega­tive parity states. In both the nuclei the j>(IJ dependence is olose to that obtained ~ the statistical model. At high excita­tion energies the results of semi-a.iorosoopio calculations oome nearer to the statistical ones. At high energies troubles with direct calculations become much more serious while the accuracy becomes worse due to insufficiently correct account of the PaUli principle because of the neglect of degeneration of certain states in spherical nuclei and ambiguity in the expression of the many-

i

-quasiparticle configuration in terms of noncolleotive phonons. Therefore one should use the direct method for 8 o:o Bn and. the statistical one for & > Bn.

The correct description of the level density provides also evidence for the fact that this model may serve as a basis for describing the highly excited state structure since the configura­tional space of the mod.el is large enough to cover the complexity of highly excited states.

4. We give a part of the results on fragmentation obtained in collaboration with Dr. MaliW ey illea.ns of an ap]H'4xilna.te -SOlrlng

of eqs. (5) and (6). The calculations are made for the states with 1( 10, 12~ in 239u. The Saxon4Yoods potential parameters, the

interaction constants and some particular features of the calcula­tions are given in ref. 15 • In studying fragmentation, similarly as in calculating the level density, we use the parameters which were fixed in the stud.y of low-lying states. Fifteen multipolari-ties AJ.l , which are given in Table I, and 10- 70 solutions of the secular equations for phonons are utilized. in the calcula­tions. The wgve function (2) is seen to have a large number of different components and therefore it can describe the complex

structure of states.

The j"AJli Table I

values and the number function (2).

of terms of the wave

---- ----- -----Number of terms of Af- th~ type _____ --------------20 22 JO Jl .32 JJ 41 4J ~ '. Q f .j_:- Q; Q;, 44 54 55 65 66 76 77 .")-,, i

--- ------j = 1, 2, ••• 10 870 9.104

j = 1,2, •••• .35 1.104 1.106

j = 1,2, ••••• 70 5.104 5.10 6

-----Finding of the solutions for eqs. (5) and (6) and calculation

of the wave function components require much electronic computer time. Up to the present time one has obtained. a total of about 100 solutions for these equations and. for the appropriate wave functions. As an example, in table 2 we give a typical state structure. The component values are calculated from the normali­zation condition for the wave function and are expressed in per­cent. This state corresponds to the 6221 +0 1 (31) + 0 1 (.31) fundamental pole of an energy 2.07 MeV. For the phonon we use the

9

Table 2

Squared components ( in percent) of the wave function (2) for the 1.9 MeV 1/2+ state in 239u

notation fl/ i.ji) and for the single-particle state the Nilsson model notation ilnz A I and iln! .-1 t • The root of eq. (5) is

lowered by 0.17 MeV respectively . to the fundamental pole. With

Configuration Components increasing excitation energy the

--------%_ roots drop down still further 620+ 6.10-4 with xe~t te the fundamental

5oll + 0, (31) 28.98 poles quasiparticle plus phonon.

761. + o, (Jl) 1.47 It is seen from table 2 that 52.4~

752t + o, (Jl) 0.15 of the state strength is conoent•

725+ + o, (55) o.o4 rated about the component corres-

6241 + t'L(44) o.o1 pending to the fundamental pole.

6221 + O,(Jl) + O,(Jl) 52.42 Next, the phonon-containing com-

7701 + o, (22) + 0,(31) 6.66 ponents entering the fundamentAl

6311 + o, (3o) + o,(31) 2.14 pole are predominant. This appe-

6o61 + o,(31) + o,(55) 1.99 o.rs to be due to the approximate

5011 + o, (20) + 0,(31) 1.3o method of solving eqs. (5) and

74Jf + 0,(22) + 0,(31) o.8a (6).

503+ + o, (22) + o,(31) 0.69 In some states the value of

6241 + 0,(31) + 0,(55) 0.44 the component corresponding to

606t + o,(Jl) + o~(55) o.Jo the fundamental pole reaohes 9~.

5031 + Q,(31) + Q,(44) 0.25 Insufficient fragmentation of

6J3+ + o, (31) + o,(31) 0.22 certain states quasiparticle plus __________ phonon and quasiparticle plus

two phonons and great fluctuatims of the strength from level to level are due to the fact that, firstly,aocount is taken of only the multipole oolleotive states, other collective vibrations, e.g.,g:l.ant resonances, are disregar­ded. Secondly, the wave function (2) has no terms containing three and more phonons, thirdly, eqs. (5) and (6) are solved approxi­mately. To get better description the approximate solutions for eqs • (5) and (6) oorrespondlng to the one-quasiparticle and quasi­particle plus phonon fundamental poles should be improved. It is also necessary to perform in the wave function (2) a summation over one-quasiparticle statesjP~

Let us consider the fragmentation of the one-quasiparticle states and look how it changes depending on the position of the single-particle level with respect to the Fermi level. We take four one-quasiparticle states with Kx. ~2+ , two of them are near the Fermi level, one state is a high particle state and the other is a deep hole state. To study the fragmentation of these states

10

it is necessary to find several thousands of solutions for eqs.(5) and (6) and their wave fUnctions. The investigations showed that the energies and wave fUnctions for the solutions corresponding to the fundamental poles quasiparticle plus phonon strongly differ in the two-phonon and one-phonon approximations, i.e.,in the oase when the terms quasipa:rt:l.cle plus phonon are taken into account or d:l.sregarded. In the one-phonon approx:l.mat:l.on the fragmentation of states quas:l.partiole plus phonon :l.s badly described, and for many solutions the component corresponding to the fundamental pole exceeds 9glj. However; the distr:l.bution of the singJ.e..:partfcle strength is nearly the same in the one- and two-phonon approximat­ions. For example, for the 63lt state at the first level in the one-phonon approx1mat:l.on the strength concentration-.:ls 90.1%, in the two-phonon one 85.~. The total 6Jlt strength at all the le­ve~of an energy up to 1.9 MeV :l.n the one-phonon approximation :l.s 9J*, :l.n the two-phonon one 92.~. For the 620+ state without the solution corresponding to the one-quasiparticle fundamental pole the total strength of the 620 state at all the levels up to an energy 1.9 MeV is 1~ in the one-phonon and 14% in the two-phonon approximations. Tberefore,to study fragmentation of one-quasi­partiole states we use the one-phonon approximation.

The calculation results are given in Fig.l • The (C)/ values are calculated from the normalization condition (3), represented as a sum over the states ly:l.ng in the 0.2 MeV energy interval, denoted as (C/'/= Z:(C)} 2 and given in percent. The excitation enere;r is

' )~ pl.otted on the abso.issa axis. The2

( C6 a1 t quantity is plotted on the left ordinate axis. The (c)') values for the 620 ~ ' 600 + and 640+ states are plotted on the right ordinate axis. The compact presentation of the data has given no possibility of following firmly the soale for some ( CJ' / values, which are therefore marked by appropriate numbers.

In the lower part of the figure we give the fragmentation of the 6Jlt single-particle state placed below the 622'.Fermi level by 0.2 MeV, for it E (6Jlt) = 0.8 MeV. At the first level the strength concentration is 90.1% , at the levels in the 1-3 MeV interval we have another 5.9~ of the strength. A total of 96.6% of the 6311 strength is exhausted up to an excitation energy 3.8 Mev the rema:l.n:l.ng J.4~ relate to higher levels. Such a situation with the main component is typical of the low-lying states of odd de­formed nuclei, ref.8 • The particle 620# state is by 2 MeV higher than the Fermi level, for 1t E (620f) =2.1 MeV. At the lowest o.s MeV level the strength conoetration is 6~, at the levels of 1-2 MeV another 16.~ of the strength is concentrated. Further the

II

( Cs31l %

,. 13

12

fl

10

9

8

7

6

5

4

J

2

I

9111

D' 8.7

0

640t

2 (~01)

6

5

+

0/o

60

.

0

Ill ~ I L I u ~ II t; ~I -, ~1 I I IO

25.7

[J ~~· L:J

600t

:c500l

5

4 J

2

o/o

..._,...- I 'O ,......, I I (CS20/ 0

0/o .5

2

1 I I I ..... ~•••'~ I I I ..__, ~ L.....l "=-1_ Jo 63H

O' I II I I r= .......... - I

0 E(6lfl) I 2 3 _l_ ! ,,MeV Fig.l ·u t-

Strength distribution for single-particle states with K =lz in 2J9u

12

I

i

strength function has a minimum, after J MeV a rise is observed. Up to 5.2 MeV excitation energy 90.~ of the 6201 state strength is exhausted. The study of the fragmentation of the 6Jl t and 620 I state near the Fermi surface has shown that their strength distri­bution has a long tail. B.r means of this tail it appears to be possible to explain the values of the s- and p-wave neutron strength functions in the range of their minima.

The particle 600 I state is by 4.J MeV higher than the Fermi level, t (6601) = 4.36 MeV. At the levels below 2 MeV there is less tlian r percent of the strengtn, in tnif lliTerva.I 2.4-3.4 MeV there is 5~ of the 6001state strength. Up to 5.2 MeV energy 64.5% of the strength of this state is exhausted. It is unclear how the remaining 35.~ is distributed. It should be noted,that, in spite of the common opinion 1the particle state strength is distributed in a nons;ymmetric way with respect to the single-particle energy. So, 7~ of the 6201 state strength is exhausted up to 2 MeV, 60% of the 600+ state strength up to 4.3 MeV and 5~ of its strength is at the levels up to 3.2 MeV. Thus, for the particle states the distribution maximum is bia~ in favour of low energies.

The 640 I hole state is by 3.04 MeV lower than the Fermi level, £ (640 0 == 3.44 MeV. At the 1/2+ levels of 1-J MeV energy there is 4~ of the 640.state strength, between J and 4 MeV the distribu­tion has a deep minimum, then a noticeable increase is observed. Up to 5.2 MeV excitation energy 72.7~ of the 6401 state strength is exhausted. The fragmentation of this state is characterized b;y strong fluctuations of the strength not onl;y from one 0.2 MeV zone to another but especially from one level to another. So, at excitation energies exceeding 4 MeV there may be levels which ha~640• components equal to 3-' $.It is unclear whether these large fluctuations are due to spe cifio features of the 640 I state of to the bole-state fragmentation.

The account of the terms quasiparticle plus two-phonons Which is taken in the study of the distribution of the one-quasiparticle st.ate strength results, firstly, in anirorease of the state density due to the appearance af the solutions corresponding to the fundamen­tal poles quasiparticle plus two phonons and due to decreasing ener­gy of the states correspoming to the poles quasiparticle plus one phonon, secondly, in an increase of the fragmentation of one quasi­particle components over the levels and in a decrease of the str~ fluctuation in the transition from one level to another.

The study of the fragmentation of the states quasiparticle plus phonon and quasiparticle plus two phonons continues. As an example, table J contains the fragmentation of six states quasiparticle plus

13

phonon over eight levels in 239u. We give only those camponents which are larger than 0.01~ • For the time being it is still difficult to formulate the particular features of the fragmentati­on of the states quasiparticle plus two phonons.

5. The model considered above is rather simple. It permits describing the energy and the structure of nonrotational states with a lower excitation energy than that at which there appear fundamental quasipartiol e plus three phonons. At excitation ener­gies, at which the number of the fundamental poles quasiparticle plus three phonons is large enough, the calculations with the wave function (2) yield the strength functions rather the structure of individual resonances.

The approximate method developed in ref. 17 is found to be ve-_ ry effective. Using it one has succeeded in obtaining approXimate solutio.ns for more- complicated oases when the wave function (2) is supplemented with the terms quasiparticle plus three phonons and when the wave function consists of one-, two-, three- and four-phonon components. Thus, there is a possibility of continuing the study of the state structure at high excitation energies and the fragmentation of components with large number of phonons.

Of great interest is the study of the widths and the fine structure of giant resonances and the coupling between these collective states with different multipolarities. The model in question and its modifications are found to be suitable for solving these problems. For example, collective states, such as giant resonances in odd deformed nuclei, can be described by

1.1 TlJ-~jD'' +f1++~Fu.i,+ n•a+-+ 'f't ( K - f:n: YITJ)(I' I 7-z '/D"' ci- Y,O' (:J._J Jz. (9)

~L. R9M3'J+ a· a· a .. J ~ 3•./J YIT '" ! I~ /J q •

In this wave function we have neglected the one-quasiparticle terms which little contribute to the wave function normalizatio.n. The probabilities of i' - r&J excitation for these states are proportional to I DY!~ I', where >lc - corresponds to the ground one-quasiparticle state. The summation over J , !• and j 5

is performed not only over collective states of the type of giant resonances with different multipolarities, but also over low-l;ring collective states. The second and third terms in eq. (9) are responsible for the width and the fine structure of the giant re­sonance of appropriate multipolarity as well as for their mutual influence. The model considered in ref. 16 can be used for the study of the structure of giant resoDaiJC es in doub17 even deformed

14

I ~ j

nuclei. It is interesting to clarify how the giant resonances affect

the process of fragmentation and thereby the structure of excited states of energies lo11er than the energy of giant resonances. It mar be hoped that our model will be effective for answering this question.

The suggested model cannot cover the whole complexity of highly excited states. structure complications are, to a large extent, due to th.e interaction of singie partLcla 111ntion with va­rious types of collective and weakly collective motions. Therefore it is necessary to account for other collective states, such as spin-multipole and Gammov-Teller states, giant resonances, etc. Searches for other, not so clear-cut, collective branches at inter­mediate and high excitation energies should be intensified.

Table J --Fragmentation of states quasiparticle plus two phonons over eight levels of 239u

State I Components in percent, Configuration I fund. Ener MeV

pole, e 1.52 2.02 2.09 2.20 2.J2 2.52 2.5J 2.54

6221 +0,(JO)+Q,(J2) 2.49 12.1 - 75.9 6Jll +Q,(Jl)+O,(Jl) 2.57 - 61.!5 - 0.92 - - - 0.16 6221 +O,(Jl)+O,(J2) 2.74 - 1.0 - 58.6 0.06 - o.ao 6221 +Q,(J2)+Q,(55) 2.90 - - - - - 79.2 0.44 20.J 6241 +Q,(Jl)+O,(J2) 2.94 - - - - o.Ja 5.75 0.97 55.1 6Jlt +OP2)+0.(J2) 2.96 o.Ja - 0.06 0.15 0.64 0.01 40.4 6.16

6. We formulate the foundations of the general semi-micro­scopic approach based on the operator form of the highly excited state wave function a:ril estimate the contribution of some simple configurations in the wave fUnctions of neutron resonances.

Within the framework of the semi-microscopic approach of the superfluid nuclear model we construct 11 •12•15 the wave function of a highly excited state which is presented in the form of an expansion in the number of quasiparticles. The state structure comp­lication with increasing excitation energy is seen from the fact that in the wave function components with over-increasing number of quasiparticle& become more and more important. In constructing such a wave function we start from the interaction Hamiltonian (1) supplemented with other residual interactions and employ the sing­le-particle states of' the average :f'i-eld potential and the mathema-

15

tical apparatus of the superconducting pairing correlation theory. The wave function of the highly excited state of an odd-A-sphe­

rical nucleus is of the form

J,/(12M)=b.AJ+ 'fl.+ TA .1. ]ltl 0 (10)

~ ~ + + • Ill + L- b

1 ( J, m, J, m, 13 mJ )J1,m, J..j,m,J.J, m, To •

J,J,Jj ff1,f11,tl71J

~ b). . · - · )' r· J. • t• J. • t• ,,; + + L_ 1 (J;'11, J.!o7 1J'fiJJo~n7vJ.in1;,4 d-j,Ft1, :ilml•iJmJ :J~!,..,f d...j~niJ· To

J,J~ JjJ.,J,· m,m~'"Jm~nl,f

This expression should be supplemented with the terms having the operators of pairing vibrational phonons which replace the opera­tors (/.1~, .1./-~,;I=o • In addition, we can introduce explicitly in (10) the operators of any phonons.

When constructing the wave function (10) we assume that the density matrix is diagonal for the ground state of the nucleus. In this representation the wave function of a highly excited state must contain thousands of different components. The use of this representation is physically justified, In the majority of oases the formation of a highly excited state occurs due to capture of a slow neutron or a high-energy r- ray by a target nucleus in the ground zero-quasiparticle or one quasiparticle state. Therefore the expansion (10) seems to be performed in the basis functions of the

target-nucleus. The operator form of the wave function is used in refs.

11•12

to express the reduced neutron, radiational and alpha widths in ).

neutron resonances in terms of the coefficients bi • In this way one formulates the problem of experimental determination of the coefficients b; • The coefficients b; can be found from the spectroscopic factors of the reactions of the type (dp) and (dt), the B decay probabilities, gamma transition probabilities between

excited states,etc. Consider, for example, the one-quasiparticle components. The

strength of each single-particle state is distributed over several levels. At low energies this fragmentation is manifested in (dp) and (dt) reactions when some levels are excited. With increasing excitation energy the level density grows, and it. is difficult to re­ool~·them experimentally. Therefore, in ref.

12 it is proposed to

determine experimentally the strength functions in (dp) reaction with momentum transfer t < 1 and compare them with the strength

functions for p-wave neutrons. In this way it is possible to study

16

1 I

~

the energy dependence of the strength function. The (dt) reaction cross sections can yield information about strength functions acco­ciated with the hole state fragmentation. It is of particular concern to clarify how the fragmentation of the hole state strength differs from that of the particle state strength. There has appea­red first theoretical and experimental investigations 20 • 21 on the study of (dp) reactions on unbound states. In ref. 22 an attempt has been made to obtain information on neutron strength functions in deformed nuclei, In tQat ~aper one has not succeededinohta1mng the values of the strength functions for fixed momentum transfers.

Let us consider, for example a reduced neutron width which can approximately be written in the form

r.. • r_• 1..)" · ·2 (11) nA = sp. f L' ( J) Uj / ,

where the function Uj indicates that the state j must he a ;

particle state. Knowing the exper:llnental values of<~~~ > averaged - z over a number of resonances it is not difficult to find I b I .

For ~ "'Bn they are found to be the following: for nuclei Ca - Ni /bl 2 ~ 10-J for Zn- Ba and certain isotopes of Au and Hg /bi 2-IO.':.to·•

- 2 J -4- - 2 -6 -7 for Pb isotopes I b I ~· 10- -10 , for deformed nuclei I b I ~ 10 -10 • Note that the values obtained in refs. 12•1 J are found to be under­estimated by one-two orders because of an incorrect interpretation of the experimental data,

We now investigate in which oases the valence neutron model is valid. The matrix element of the El transition from the highly excited 'state described by the wave function (10) to the one-quasi­particle state clj;m1 'II. is of the form

i;-11•1

.M (Et,· rA --4 mf)= (-'jy (J~ fl?; I-M/II)b/ l~jj <.i lr( E 1)/ )1 >-

-J:s. J. (- t).i,j,-(j,m,j,m,ltl)b;( J1m.;j,m,)zm,)7Jj,j, <i,lr{EI) U. > +- "·,

J, lz

(12)

where <j I r(Ef) jj'>- is the s1ngl..-oarticle matrix element of the E1 transition. The first term in {12) describes the El transition in the valence neutron model, the remaining terms describe the E1 transitions from the three-quasiparticle components of the neut­

ron resonance wave function. Consider the El transitions from p-wave neutron resonances to

the low-J.ying states in 91ao which illustrate the valence neutron model 2J • !'ig.2 gives the 9luo low-J.ying states with the indica­tion of the one-quasiparticle component contribution and the three­quasiparticle states lying, according to the calculations, in the

17

7,5-B.SMeV

P,!z \'lz-

(693

1491

0

7.5-8.5 MeV interval from which a,

1z'S,13•P,1, the I:l transitions can occur tc

--d. •h •9

one-quasiparticle states, As the S/z "lz 9/z --a •a •P calculations shew, the subshells S/z Slz 3/z

--a~lz•dsJ/fslz JP,1!and JP.l/zare relative to

3/~ Pl/z

1/z • a.:sz Sr;2

.5/z" 0.12 d,-12

3/2' O.Sf d31z

t/z• 0.64 5,;2

5/z • D. 84 d.s-;z

5.06 MeV and 6,81 UeV energies. They are distributed over a large number of levels in a wide energy interval. Due to this fragmenta-tion the wave functions of p-wa­ve neutron resonances near 8n ~

= 8,067 MeV contain noticeable one-quasiparticle components J PrJz or J P31z , which are res­ponsible for the El transitions in the valence neutron model. In addition, the El transitions proceed from three-quasiparticle components which are due to fragmentation of three-quasipart­icle states ( Fig.2). These com­ponents lead to the violation

Fig.2 • ZI-transitions from one-and three-quasiparticle

of the valence neutron model in 93Mo for the I 71: J/2- resonances which is confirmed by experimen­tal data. It should be noted that

components of p-wave neutron the violation is also due to resonances to low-lying states three-quasiparticle component in 9JMo admixtures in low-lying states.

The calculations show that in the

case of 99uo near Bn there are no three-quasiparticle states from which the El transitions occur to the subshells 3SJ/z J 2Ci,-;2

and 2d~12 • Therefore in this case the valence neutron model is expected to show nore clearly its validity.

According to ref. 24 this model is quite valid for the El transitions in ( rn ) reaction on 36 J/2- resonances in the (5-225) KeV interval higher than Bn = 7. 22 MeV in 91 zr. A strong correlation (jD =0.59) between neutron and radiational widths is obtained. According to our calculations in 91zr, the 2P.l/z subshell has an energy of about 7 MeV, and in the 6-8 MeV interval there is not a single three-quasiparticle state to which the El transition from the ground state could occur. There­fore in this case the valence neutron model is expected to work

II

( ~

well. However, in the 8.1.-3.4 !,leV interval we have the follow"inr; three-quasiparticle states:

2ri5lz r 2d;;z + f h;z ,2ds-;z r 2d;72. 2.1J;.!,2d.fi?.rJS'/2 #Lfl'fz,ld,/2 '6

/hll:'%;,}0 Which the El transitions from the !ct<1., state can !)roceed. Therefore with increasing excitation energy the valence neutron model in f,J"n) reaction in 91zr is expected to be less profitable.

In some cases experimental data on radiational widths permit obtaining information about certain components of the wave function (lO). Of most conveni~ a,re the racti&n& &.f the type ( r,.,), sine e the known structure of the initial state allows one +o ~ :bxU­

vidual components in the excited state. In some cases it is possib­le to obtain infonnation about three-quasiparticle and five-quasi­particle components of the highly excited state wave functions. In this respect the most favourable is the study of the El transitions in 177Lu from neutron resonances w1 th Ill'= lJ/2- and 15/2- to three-quasiparticle states,

Sor.1e simple configurations vrere estimated in ref. 25 from the experimental data on ( fl/' ) reactions by calculating the EI tran­sitions from capture states to the low-lyine ones. In these papers the low-lying states in 57Fe, 59

t63ui and in nuclei with ir=28 and

82 are described taking into account quadrupole phonon admixtures. It is shown that the components quasiparticle plus phonon in the WaYe functions of the capture states are very important in the EI transitions to the low-lying states.

It is very intersting to clarify the problem about rotation and the nuclear shape in highly excited states. On the basis of a large amount of experimental information ( strength function be­haviour for s- and p-wave neutrons, probabilities for ~ and J" transitions from highly excited states, spliting of the EI giant resonance and others) it is possible to canol ude that the shape of spherical and deformed nuclei for the majority of states does not change essentially with increasing excitation energy. The problem of the shape of the excited states of transition nuclei is more complicated. Undoubtedly of importance are direct measu­rements of the nuclear shape in highly excited states. Tn this connection, the suggestion of Ostanevich 26 to measure isomer shifts of neutron resonances is of great interest.

To clarify the particular features of nuclear rotation in highly excited states in ref, 27 it is proposed to investigate whether the K forbiddenness in the r transitions from neut­ron resonances to low-lying states takes place or not. The study28

of the EI transitions in l77Lu from 1/C = lJ/2- and 15/2- resonan-176 ces produced as a result Of 8-waYe neutron capture af Lu has not

19

shown any large enhancement of the EI transitions to low-lying levels with large K compared with the transitions to levels with small /( • Hence it" may be concluded that K is not a good quan­tum number in highly excited states. It should be noted that consideration of rotational motion in ref. 19 has led to an increa­se of the high-spin level density at ~ = Bn by several times. This points out that rotational motion is important in highly excited states of deformed nuclei. However, it seems to be imposs~ le to distinguish it explicitly among other forms of nuclear motio~

7. At excitation energies close to the neutron binding energy I3n and higher the wave function (10) contains thousands of vario­us components. In many quasiparticle components of the wave funct­ions the quasiparticles are distributed over bound single-particle average field levels, and not a single nuclear of this configura­tion is able to leave the nucleus. The few-quasiparticle components are responsible for the nucleon emission. The many-quasiparticle components of the wave function (10) correspond, to some extent, to the quasi-bound state discussed in ref. 4 •

Such wave functions possess the properties of the compound states introduced by li.Bohr. Our treatment of the highly excited state differs from the Bohr conception of compound state. We are based on the introduction of the average field and residual interac­tions and use the representation in which the wave function of a highly excited state is a many-component one. In this way it is possible to understand all the effects which are interpreted by means of the compound state. However, with such an approach the question does not arise as to how such a complicated state is for­med dynamically from a simple state due to nucleon or ~-ray ca~.

29 In ref. one puts the question as to whether all the compo-nents of the highly excited state wave function are small or among them there are relatively large ones. The ways of experimental discovery of large many-quasiparticle components of the wave funct­ions of neutron resonances are discussed in refs.lO,lJ • At presett

the most available way of clarifying the role of the many-quasi­particle components is the study of El-, 111- and E2-transitions from neutron resonances to states of an energy by (1.0-1.5) MeV lower than their energy. Possibly the probabilities for these transitions can be estimated from the study of the subsequent alpha-decay of

JO the excited state ( e.g. 1in ref. ), fission or neutrc.n emission.

Observation of the f- transition cascades, the reduced probabili­ties of which are close to the single-particle ones,provides evi­dence for the existence of large many-quasiparticle components in the wave functions of neutron resonances and in states of inter-

20

... ( ~

mediate excitation energy. Information about the magnitude of in­dividual four- and six-quasiparticle components can be extracted from the study of '{"- transitions from neutron resonances to states of intermediate excitation energy. F0r example, in ref. 31 ~- tran­sitions in

58Fe, obtained after capture of a thermal neutron, to

states of an energy higher than 5 MeV are studied. It should be noted that the presentation of the wave funct­

ion as simple and more complicated parts is widely used in trea­ting excitation reactions of highly excited states. However, the mathemat1cal ~thod of a gradual introducnon of simple and more complicated components should not be understood literally. It is impossible to assign to this method the physical meaning of tran­sitions from simple to complicated configurations.

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don Press, Oxford, 1968. J) I.M.Frank, nuclear structure study with neutrons, ed.by Er8

and Szttos, p.J, P1enum Press, 1974; I.Bergqvist, Nuclear structure study with neutrons, ed. by Er8 and Szttcs,p.J, Plenum Press, 1974.

4) V,Gilet, Nuclear Structure Dubna Symp. 1968,p.27l,IAEA, Vienna, 1968; N.Cindro, 6th Intern.Summer School on Nuclear Physics, P• 600 ,1974; C. Bloch, Proc. of the Enrico Fermi School, Varenna, Session J6 ( Academic Press, New York, 1966).

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6) V.G.Soloviev, Izv.Akad.Nauk SSSR (ser.fiz.) 1§ (1974) 1580 7) V.G.Soloviev, Nucl,Structure Dubna Symp. 1968, p,lOl, IAEA,

Vienna, 1968.

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Er!J and Szttcs, p.85, Plenum Press, 1974 .

21

14) V.G.So1ov1ev, Izv.Akad.NaUk SSSR, (ser.fiz.) 22 (1971) 666. 15) V.G.So1ov1ev, L.A.Malov, Nuc1.PbYS• ~(1972) 433. 16) V.G.Soloviev, Teor.Mata.Fiz. !1 (1973) 90; A.I.Vdo~, V.G.So-

10viev, Tear. Math.Fiz. ~ (1974) 275; G.Kyrchev, V.G.Soloviev, Preprint JINR E4-7764 (1974).

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Yad.Fiz. ~ (1974) 516; V.G.So1ov1ev, Ch.Stoyanov,A.I.Vdovin, Nuc1.PbYS• ~ (1974) 4U.

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Phys.Rev.Lett. g§ (1971) 1118; O.A.Wasson, G.C.S1angher, Phys.Rev. ~ (1973) 297.

24) R.E.Toohey, H.E.Jackson, Phys.Rev. ~ (1974) 346· 25) V.A.Knatko, E.A.Rudak, Yad.Fiz. !2 (1971) 521; Yad.Fiz. !2

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A215 (1973) 45.

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4.85

12.59

9.96

6.36

9.54

6.78

5.00

21.00

8.76

10.53